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In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surface materials undergoing deformation and phase segregation. To this end, we devise a principle of virtual powers with a bulk-surface dynamics, which is postulated on a material body P where the boundary ∂ P may lose smoothness, that is, the normal field may be discontinuous on an edge ∂ 2 P. The final set of equations somewhat resemble the Navier–Stokes–Cahn–Hilliard equation for the bulk and the surface. Aside from the systematical treatment based on a specialized version of the virtual power principle and free-energy imbalances for bulk-surface theories, we consider two additional ingredients: an explicit dependency of the apparent surface density on the surface thickness and mixed boundary conditions for the velocity, chemical potential, and microstructure. • A mathematical framework for the mechanical interplay of bulk-surface materials. • Explicit dependency of the apparent surface density on the surface thickness. • A principle of virtual powers with bulk-surface dynamics. • Bulk-surface Lyapunov decay relations. [ABSTRACT FROM AUTHOR]